Who is the state head, of this kind of territory?
The state head is the current state head of the United States, in this case Barack Obama
Why can't they vote for the United States President inside of their territory?
As the president is elected by the electoral college, which in turn is composed of a number of electors. The number of electors in each state is equal to the number of members of Congress to which the state is entitled. As Puerto Rico (as does Guam too) isn't represented in Congress, they have no electors and can thus not influence the election. So the question should probably be "Why is Puerto Rico not represented in Congress?". To that question one could answer that the territory of Puerto Rico is simply the property of the U.S. and not an independent body, thus it is up to the federal government if they grant any rights to the structure of Puerto Rico (as has been done in 1950).
How is their budget assigned?
The contribute and benefit to/from the federal budget of the U.S.A. and have their own "state" budget similar to the 50 states.
Do they receive federal help in cause of natural disaster?
Are they allowed to receive help in case of economic troubles?
In 1992 then president Bush wrote a letter to all the federal agencies stating that they should treat the same way than one of the 50 states. As they contribute to the federal budget they do also receive help from it.
The Puerto Ricans are also contributing to the social systems and benefit in case of need from them.
Which are the differences between this kind of territory and a normal state of the Union?
Puerto Ricans only benefit of the basic rights and have no representation in the US houses. They do pay all federal taxes but may be subject to local taxation too. They do also serve in the U.S. military branches. In principle the only distinction is the rights they hold vis-à-vis the 50 states citizens.
And more important in what part of the Constitution this system is allowed, and who invented it?
Constitutionally, Puerto Rico is subject to the Congress's plenary powers under the territorial clause of Article IV, sec. 3, of the U.S. Constitution.
"The Congress shall have Power to dispose of and make all needful Rules and Regulations respecting the Territory or other Property belonging to the United States ..."
So nobody except the founding fathers invented it.